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## Q. 12.5

A plane layer of gray medium with thickness D and constant properties is initially at uniform temperature $T_0$. It has attenuation coefficient β, density ρ, and specific heat $c_υ.$ At time t = 0 the layer is subjected to a very cold environment. Consider only radiation transfer and obtain the transient temperature of the layer by using the emission approximation.

## Verified Solution

From the results of Example 12.4, the instantaneous transient heat flux emerging from both boundaries of the layer is $q(t=)4\beta \sigma T^4(t)D$.The energy equation for the layer then becomes $ρc_υdT /dt=-4\beta \sigma T^4(t),$or, in dimensionless form,$d\vartheta /\overline{dt} =-4\vartheta ^4(\overline{t} )$ where $ϑ=T/T_0$ and $\overline{t}=(\beta \sigma T_0^3/\rho c_v)t.$ Integrating with the condition that ϑ = 1 at $\overline{t}=0$ gives the transient uniform temperature throughout the layer for the emission approximation as

$\vartheta (\overline{t} )=\frac{1}{(1+2\overline{t})^{1/3}}$